Timeline for Fourier transforms of compactly supported functions
Current License: CC BY-SA 2.5
8 events
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| Jul 7, 2010 at 17:32 | answer | added | user7361 | timeline score: 10 | |
| Jul 2, 2010 at 1:57 | comment | added | Phil Isett | Thanks for these replies. Maybe I should clarify: I'm not exactly looking for a proof of a fact for more general topological groups. (I'm really grateful for the example, though.) I just thought the key difference between the p-adics and Euclidean space in this regard has to do with the fact that convolutions (of, say, non-negative, compactly supported functions) really do expand support on R^n, but don't need to on the p-adics. I'd like to see a proof that highlights that feature of Euclidean space if there is one. | |
| Jul 1, 2010 at 13:49 | comment | added | BS. | Yemon, I don't think this to be true : the inverse limit of $2\times : \mathbb T \to \mathbb T$ (a so called solenoid) seems to be connected. See en.wikipedia.org/wiki/Solenoid_(mathematics). | |
| Jun 30, 2010 at 23:38 | history | edited | Phil Isett | CC BY-SA 2.5 |
Clarified question
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| Jun 30, 2010 at 6:54 | comment | added | Yemon Choi | By the way, there ought to be a classification of connected locally compact abelian groups in Hewitt & Ross volume 1 (though that may not be the most effective or convenient source). IIRC, they are all of the form ${\mathbb R}^n \times {\mathbb T}^k$. | |
| Jun 30, 2010 at 6:52 | comment | added | Yemon Choi | Scott, I don't think that's what's being asked? As I understand it, the question is something like the following: is there a "connectedness-driven proof" that, when G is abelian connected and locally compact, and f is a function vanishing on an open subset of G, then the FT of f cannot have compact support. | |
| Jun 30, 2010 at 2:55 | comment | added | S. Carnahan♦ | I don't think connectedness alone kills compact support in frequency space, since you can have compactly supported functions on $\mathbb{Z}$ or $S^1$ (or products of these) with compactly supported Fourier transforms. | |
| Jun 30, 2010 at 0:43 | history | asked | Phil Isett | CC BY-SA 2.5 |